Question

Formic acid \(\left(\mathrm{HCO}_{2} \mathrm{H}\right)\) is a monoprotic acid that ionizes only partially in aqueous solutions. A \(0.10 M\) formic acid solution is \(4.2 \%\) ionized. Assuming that the molarity and molality of the solution are the same, calculate the freezing point and the boiling point of \(0.10 M\) formic acid.

Short answer

The freezing point of the \(0.10 M\) formic acid solution is approximately \(-0.194\) °C and the boiling point is approximately \(100.053\) °C.

Step-by-step solution

Calculate the Van't Hoff Factor

To calculate the Van't Hoff factor (i), we need to consider the percentage of ionization given. Since formic acid is a monoprotic acid, it can ionize to form one hydronium ion (H+) and one formate ion (HCO2-). Therefore, the Van't Hoff factor is calculated as follows: i = 1 + α Where α is the degree of ionization, which is given as 4.2%. We'll convert the percentage to a fractional value. α = 0.042 So, i = 1 + 0.042 = 1.042

Calculate the molality from molarity

Since we are given that the molarity and molality of the solution are equal, we can use the given molarity value as the molality: molality = 0.10 mol/kg

Calculate the freezing point depression

To calculate the freezing point depression, we will use the formula: ΔT_f = i * K_f * molality Where ΔT_f is the freezing point depression, K_f is the cryoscopic constant of water (1.86 °C kg/mol), and molality from step 2. ΔT_f = 1.042 * 1.860 * 0.10 ΔT_f = 0.19398 °C Now, we can calculate the freezing point of the formic acid solution by subtracting the freezing point depression from the freezing point of pure water (0 °C): Freezing point = 0 - 0.19398 = -0.19398 °C

Calculate the boiling point elevation

To calculate the boiling point elevation, we will use the formula: ΔT_b = i * K_b * molality Where ΔT_b is the boiling point elevation, K_b is the ebullioscopic constant of water (0.512 °C kg/mol), and molality from step 2. ΔT_b = 1.042 * 0.512 * 0.10 ΔT_b = 0.053344 °C Now, we can calculate the boiling point of the formic acid solution by adding the boiling point elevation to the boiling point of pure water (100 °C): Boiling point = 100 + 0.0533 = 100.0533 °C So, the freezing point of the 0.10 M formic acid solution is approximately -0.194 °C and the boiling point is approximately 100.053 °C.

Key concepts

Freezing Point Depression

When a solute, such as formic acid, is dissolved in a solvent like water, the freezing point of the resulting solution is lower than that of the pure solvent. This phenomenon is called freezing point depression. It occurs because the dissolved particles interfere with the crystal formation of the solvent, making it harder for the liquid to freeze.
To calculate freezing point depression, we use the formula:

• \( \Delta T_f = i \times K_f \times \text{molality} \)

In this formula, \( i \) is the Van't Hoff factor, \( K_f \) is the cryoscopic constant (1.86 °C kg/mol for water), and the molality is the concentration of the solution in mol/kg, which, in this case, is equal to the molarity (0.10 mol/kg).
This means the freezing point of the 0.10 M formic acid solution is slightly below water's freezing point, calculated at approximately -0.194 °C.
It's important to remember that freezing point depression is proportional to the number of solute particles in the solution.

Boiling Point Elevation

Boiling point elevation is the increase in the boiling point of a pure solvent when a solute is dissolved in it. This occurs because the addition of solute particles requires additional energy for the solvent molecules to escape into the vapor phase. As a result, the boiling point of the solution increases.
The formula to calculate boiling point elevation is:

• \( \Delta T_b = i \times K_b \times \text{molality} \)

Here, \( i \) again represents the Van't Hoff factor, \( K_b \) is the ebullioscopic constant (0.512 °C kg/mol for water), and the molality is 0.10 mol/kg.
For the 0.10 M formic acid solution, the boiling point elevation calculates to about 0.053 °C. Adding this to the boiling point of pure water (100 °C), we get a boiling point of approximately 100.053 °C for the solution.
Understanding this concept helps explain why adding salt to water increases the temperature needed to reach a boil.

Van't Hoff Factor

The Van't Hoff factor, represented as \( i \), is crucial in determining the extent of colligative properties like freezing point depression and boiling point elevation. It indicates the number of particles the solute dissociates into in a solution.
For formic acid, which is a monoprotic acid, the Van't Hoff factor helps us acknowledge partial ionization. Given a 4.2% ionization, the formula \( i = 1 + \alpha \) is used, where \( \alpha \) is the degree of ionization (0.042 in fractional form).
Calculating the Van't Hoff factor as \( i = 1.042 \) shows that not all molecules ionize; around 4.2% ionize into separate ions.
This factor differs for fully ionizing solutes, such as salts like NaCl, while non-ionizing solutes, like sugar, have a factor of 1. Understanding \( i \) ensures accuracy when calculating colligative properties for solutions.